On optional stopping of some exponential martingales for Lévy processes
with or without reflection
Sören Asmussen and Offer Kella
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
1999
ISSN 14039338

Abstract:

Kella Whitt (1992) introduced a martingale $\{M_t\}$ for processes of the
form $Z_t=X_t+Y_t$ where $\{X_t\}$ is a L\'evy process and $Y_t$ satisfies
certain

regularity conditions. In particular this provides a martingale for the case
where $Y_t=L_t$ where $L_t$ is the local time at zero of the corresponding
reflected L\'evy process. In this case $\{M_t\}$ involves, among others,
the L\'evy exponent $\varphi(\alpha)$ and $L_t$. In this paper, conditions
for optional stopping of $\{M_t\}$ at $\tau$ are given. The conditions depend
on the signs of $\alpha$ and $\varphi(\alpha)$. In some cases optional stopping
is always permissible. In others,

the conditions involve the well known necessary and sufficient condition
for optional stopping of the Wald martingale $\{e^{\alpha X_tt\phi(\alpha)}\}$,
namely that $\tilde{P}(\tau<\infty) $ $=1$ where $\tilde{P}$ corresponds
to a suitable exponentially tilted L\'evy process.


Key words:

exponential change of measure, Lévy process, local time, stopping
time, Wald martingale
